The present invention relates to a simulation method of a breakdown, for simulating a phenomenon of impact ionization, which occurs in series.
In a conventional method, as shown by a dotted line D of FIG. 3, an impact ionization coefficient xcex1(E) is calculated from a magnitude of an electric field. Here, this impact ionization coefficient xcex1(E) is a coefficient on the assumption that a carrier is instantaneously accelerated. In the conventional method, by using this impact ionization coefficient, generation GII, of a carrier is calculated in accordance with an equation 6 described below (by means of an ISE manual):
GII=xcex1nxc2x7nxc2x7xcexdn+xcex1pnxc2x7xcexdp xe2x80x83xe2x80x83[Eq. 6]
And, the generation of a carrier is calculated from the calculated impact ionization coefficient, and the generated carrier is combined with an original electric current, and the current increases, and a calculation of a phenomenon in which the impact ionization is also caused by the increased electric current is automatically obtained by calculating the drift diffusion model. Here, in FIG. 3, a solid line E indicates electric field strength, and a broken line F indicates an impact ionization coefficient in case that a process in which a carrier is accelerated is taken into account.
In the above-described conventional method, the electric field strength and the impact ionization coefficient are made to be a one-to-one function, and accordingly, if an electric field exists, a carrier is fully accelerated in the field.
Accordingly, compared with an actual fact, in the calculation, the impact ionization occurs too much. Such a task occurs because, as mentioned above, the electric field strength and the impact ionization coefficient are made to be a one-to-one function.
In the conventional model, in case that an electric field exists, a carrier is fully accelerated in the field and impact ionization occurs. However, actually, for accelerating a carrier, it is necessary that the carrier runs quite a long distance.
Accordingly, in the conventional model, since the impact ionization occurs greater than a reality, the calculation is not conducted correctly. Also, due to that, a breakdown voltage becomes to be smaller than a reality. As a result, in the conventional method, the impact ionization (or the breakdown) cannot be estimated correctly.
The present invention is made to solve the above-mentioned problems.
Moreover, the objective is to provide a simulation method of a breakdown, in which the impact ionization (or the break down) cannot be estimated correctly.
In accordance with the present invention, in a simulation method of a breakdown, an electric field distribution is calculated by means of a drift diffusion model, and a one-dimensional electric field distribution is taken out from a result of the calculation, and the electric field distribution which was taken out is provided to a one-dimensional Monte Carlo device simulation to calculate a distribution of impact ionization coefficients, and Gn(x) is calculated by using an equation 1 to an equation 5 which are described in the above-described scope of claim for patent, and as a result of this calculation, it is determined that a breakdown does not occur in case that, as n becomes to be large, Gn(x) converges to a constant value, and final electric current density is calculated, and it is determined that a breakdown occurs in case that Gn(x) becomes to be large as n increases.                                           G            1                    ⁢                      xe2x80x83                    ⁢                      (            x            )                          =                                            1              q                        ⁢                          xe2x80x83                        ⁢                                          i                n                0                            ·                              α                                  n                  ,                                      (                                          x                      0                                        )                                                                        ⁢                          xe2x80x83                        ⁢                          (              x              )                                +                                    1              q                        ⁢                          xe2x80x83                        ⁢                                          i                p                0                            ·                              α                                  p                  ,                                      (                                          x                      ω                                        )                                                                        ⁢                          xe2x80x83                        ⁢                          (              x              )                                +                                    ∫                              x                0                                            x                ω                                      ⁢                                          G                0                            ⁢                              xe2x80x83                            ⁢                              (                                  x                  xe2x80x2                                )                            ⁢                              xe2x80x83                            ⁢                              α                                  n                  ,                                      (                                          x                      xe2x80x2                                        )                                                              ⁢                              xe2x80x83                            ⁢                              (                x                )                            ⁢                              xe2x80x83                            ⁢                              ⅆ                                  x                  xe2x80x2                                                              +                                    ∫                              x                0                                            x                ω                                      ⁢                                          G                0                            ⁢                              xe2x80x83                            ⁢                              (                                  x                  xe2x80x2                                )                            ⁢                              xe2x80x83                            ⁢                              α                                  p                  ,                                      (                                          x                      xe2x80x2                                        )                                                              ⁢                              xe2x80x83                            ⁢                              (                x                )                            ⁢                              xe2x80x83                            ⁢                              ⅆ                                  x                  xe2x80x2                                                                                        [                  Eq          .                      xe2x80x83                    ⁢          1                ]                                                      G            2                    ⁢                      xe2x80x83                    ⁢                      (            x            )                          =                                            ∫                              x                0                                            x                ω                                      ⁢                                          G                1                            ⁢                              xe2x80x83                            ⁢                              (                                  x                  xe2x80x2                                )                            ⁢                              xe2x80x83                            ⁢                              α                                  n                  ,                                      (                                          x                      xe2x80x2                                        )                                                              ⁢                              xe2x80x83                            ⁢                              (                x                )                            ⁢                              xe2x80x83                            ⁢                              ⅆ                                  x                  xe2x80x2                                                              +                                    ∫                              x                0                                            x                ω                                      ⁢                                          G                1                            ⁢                              xe2x80x83                            ⁢                              (                                  x                  xe2x80x2                                )                            ⁢                              xe2x80x83                            ⁢                              α                                  p                  ,                                      (                                          x                      xe2x80x2                                        )                                                              ⁢                              xe2x80x83                            ⁢                              (                x                )                            ⁢                              xe2x80x83                            ⁢                              ⅆ                                  x                  xe2x80x2                                                                                        [                  Eq          .                      xe2x80x83                    ⁢          2                ]                                                      G                          n              +              1                                ⁢                      xe2x80x83                    ⁢                      (            x            )                          =                                            ∫                              x                0                                            x                ω                                      ⁢                                          G                n                            ⁢                              xe2x80x83                            ⁢                              (                                  x                  xe2x80x2                                )                            ⁢                              xe2x80x83                            ⁢                              α                                  n                  ,                                      (                                          x                      xe2x80x2                                        )                                                              ⁢                              xe2x80x83                            ⁢                              (                x                )                            ⁢                              xe2x80x83                            ⁢                              ⅆ                                  x                  xe2x80x2                                                              +                                    ∫                              x                0                                            x                ω                                      ⁢                                          G                n                            ⁢                              xe2x80x83                            ⁢                              (                                  x                  xe2x80x2                                )                            ⁢                              xe2x80x83                            ⁢                              α                                  p                  ,                                      (                                          x                      xe2x80x2                                        )                                                              ⁢                              xe2x80x83                            ⁢                              (                x                )                            ⁢                              xe2x80x83                            ⁢                              ⅆ                                  x                  xe2x80x2                                                                                        [                  Eq          .                      xe2x80x83                    ⁢          3                ]                                          ∫                      x            0                                x            ω                          ⁢                              G            n                    ⁢                      xe2x80x83                    ⁢                      (                          x              xe2x80x2                        )                    ⁢                      xe2x80x83                    ⁢                      α                          n              ,                              (                                  x                  xe2x80x2                                )                                              ⁢                      xe2x80x83                    ⁢                      (            x            )                    ⁢                      xe2x80x83                    ⁢                      ⅆ                          x              xe2x80x2                                                          [                  Eq          .                      xe2x80x83                    ⁢          4                ]                                          ∑                      i            =            1                    N                ⁢                  xe2x80x83                ⁢                              G            n                    ⁢                      xe2x80x83                    ⁢                      (                          x              i                        )                    ⁢                      xe2x80x83                    ⁢                      α                          n              ,                              (                                  x                  i                                )                                              ⁢                      xe2x80x83                    ⁢                      (            x            )                    ⁢                      xe2x80x83                    ⁢          Δ          ⁢                      xe2x80x83                    ⁢                      x            i                                              [                  Eq          .                      xe2x80x83                    ⁢          5                ]            
Here, the above-described electric field distribution is calculated by conducting a simulation by means of the above-described drift diffusion model, in which an inverse bias is applied to a semiconductor element including an element isolating region.
Also, the above-described semiconductor element has a device structure including a two-dimensional diode structure, for example.
In addition, the above-described semiconductor element has a device structure including a three-dimensional diode structure, for example.
Furthermore, the above-described element isolating region may be formed of a LOCOS (Local Oxidation of Silicon).
Also, the above-described one-dimensional electric field distribution is taken out by cutting out a part in an electric field direction in one dimension, which has the largest electric field out of the calculated electric field distribution.
In this case, the above-described part which has the largest electric field is, for example, an end portion of the element isolating region formed of the above-described LOCOS.